which governs the hydrodynamic equilibrium of the system.
In particular, the quantum corrected metric (REF ) is still of the Kerr-Schild form [1]},
and the effective fluid is anisotropic with \(p^{\rm eff}_t\ne p^{\rm eff}_r\) .
We can then note that the metric (REF ) can be formally thought as
the coupling [2]}, [3]} of two mass functions in a Kerr-Schild spacetime,
namely
\(\tilde{m}=m_{\rm RN}+m_{\rm q}\ ,\) 
